Sum Rule of the Hall Conductance in Random Quantum Phase Transition

The Hall conductance $\sigma_{xy}$ of two-dimensional {\it lattice} electrons with random potential is investigated. The change of $\sigma_{xy}$ due to randomness is focused on. It is a quantum phase transition where the {\it sum rule} of $\sigma_{xy}$ plays an important role. By the {\it string} (anyon) gauge, numerical study becomes possible in sufficiently weak magnetic field regime which is essential to discuss the floating scenario in the continuum model. Topological objects in the Bloch wavefunctions, charged vortices, are obtained explicitly. The anomalous plateau transitions ($\Delta \sigma_{xy}= 2,3,... >1$) and the trajectory of delocalized states are discussed.